By the bisector theorem we have
and K belongs to the Apollonius circle that is the locus of points P such that
. Such a circle goes through the feet of the angle bisectors (the internal and the external) from C. If ABC is an acute-angled triangle, such a circle intersects the BC segment only at C, hence there is no solution.
On the other hand, if we allow
to be greater than
, we have a solution at